Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.5%

Time: 12.7s

Alternatives: 10

Speedup: 45.9×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Initial Program: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Alternative 1: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{normAngle}{\sin normAngle}\\ \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.5 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot u\right) - t\_0 \cdot \cos normAngle, t\_0 \cdot n1\_i\right), u, n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ normAngle (sin normAngle))))
   (fma
    (fma
     n0_i
     (- (* -0.5 (* (* normAngle normAngle) u)) (* t_0 (cos normAngle)))
     (* t_0 n1_i))
    u
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = normAngle / sinf(normAngle);
	return fmaf(fmaf(n0_i, ((-0.5f * ((normAngle * normAngle) * u)) - (t_0 * cosf(normAngle))), (t_0 * n1_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(normAngle / sin(normAngle))
	return fma(fma(n0_i, Float32(Float32(Float32(-0.5) * Float32(Float32(normAngle * normAngle) * u)) - Float32(t_0 * cos(normAngle))), Float32(t_0 * n1_i)), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Final simplification 99.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.5 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot u\right) - \frac{normAngle}{\sin normAngle} \cdot \cos normAngle, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right) \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455 \cdot n1\_i, normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right)\right), normAngle \cdot normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (fma
    (fma
     (* 0.00205026455026455 n1_i)
     (* normAngle normAngle)
     (fma n1_i 0.019444444444444445 (* 0.022222222222222223 n0_i)))
    (* normAngle normAngle)
    (fma 0.16666666666666666 n1_i (* (fma u -0.5 0.3333333333333333) n0_i)))
   (* normAngle normAngle)
   (- n1_i n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf(fmaf(fmaf((0.00205026455026455f * n1_i), (normAngle * normAngle), fmaf(n1_i, 0.019444444444444445f, (0.022222222222222223f * n0_i))), (normAngle * normAngle), fmaf(0.16666666666666666f, n1_i, (fmaf(u, -0.5f, 0.3333333333333333f) * n0_i))), (normAngle * normAngle), (n1_i - n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(fma(fma(Float32(Float32(0.00205026455026455) * n1_i), Float32(normAngle * normAngle), fma(n1_i, Float32(0.019444444444444445), Float32(Float32(0.022222222222222223) * n0_i))), Float32(normAngle * normAngle), fma(Float32(0.16666666666666666), n1_i, Float32(fma(u, Float32(-0.5), Float32(0.3333333333333333)) * n0_i))), Float32(normAngle * normAngle), Float32(n1_i - n0_i)), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{45} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{2}{945} \cdot n0\_i - \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right), u, n0\_i\right) \]

8. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165 \cdot n0\_i - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right)\right), normAngle \cdot normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right), normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, \frac{7}{360}, \frac{1}{45} \cdot n0\_i\right)\right), normAngle \cdot normAngle, \mathsf{fma}\left(\frac{1}{6}, n1\_i, \mathsf{fma}\left(u, \frac{-1}{2}, \frac{1}{3}\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455 \cdot n1\_i, normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right)\right), normAngle \cdot normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right) \cdot u\right) \cdot normAngle, normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right) \cdot u\right), normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (*
    (* (fma n1_i 0.019444444444444445 (* 0.022222222222222223 n0_i)) u)
    normAngle)
   normAngle
   (*
    (fma 0.16666666666666666 n1_i (* (fma u -0.5 0.3333333333333333) n0_i))
    u))
  (* normAngle normAngle)
  (fma (- n1_i n0_i) u n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf(((fmaf(n1_i, 0.019444444444444445f, (0.022222222222222223f * n0_i)) * u) * normAngle), normAngle, (fmaf(0.16666666666666666f, n1_i, (fmaf(u, -0.5f, 0.3333333333333333f) * n0_i)) * u)), (normAngle * normAngle), fmaf((n1_i - n0_i), u, n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(Float32(Float32(fma(n1_i, Float32(0.019444444444444445), Float32(Float32(0.022222222222222223) * n0_i)) * u) * normAngle), normAngle, Float32(fma(Float32(0.16666666666666666), n1_i, Float32(fma(u, Float32(-0.5), Float32(0.3333333333333333)) * n0_i)) * u)), Float32(normAngle * normAngle), fma(Float32(n1_i - n0_i), u, n0_i))
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right)} \]

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right) \cdot u\right) \cdot normAngle, normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right) \cdot u\right), \color{blue}{normAngle \cdot normAngle}, \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right) \cdot normAngle, normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (fma
    (* (fma n1_i 0.019444444444444445 (* 0.022222222222222223 n0_i)) normAngle)
    normAngle
    (fma 0.16666666666666666 n1_i (* (fma u -0.5 0.3333333333333333) n0_i)))
   (* normAngle normAngle)
   (- n1_i n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf(fmaf((fmaf(n1_i, 0.019444444444444445f, (0.022222222222222223f * n0_i)) * normAngle), normAngle, fmaf(0.16666666666666666f, n1_i, (fmaf(u, -0.5f, 0.3333333333333333f) * n0_i))), (normAngle * normAngle), (n1_i - n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(fma(Float32(fma(n1_i, Float32(0.019444444444444445), Float32(Float32(0.022222222222222223) * n0_i)) * normAngle), normAngle, fma(Float32(0.16666666666666666), n1_i, Float32(fma(u, Float32(-0.5), Float32(0.3333333333333333)) * n0_i))), Float32(normAngle * normAngle), Float32(n1_i - n0_i)), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right), u, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n1\_i, 0.019444444444444445, 0.022222222222222223 \cdot n0\_i\right) \cdot normAngle, normAngle, \mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \]
9. Add Preprocessing

Alternative 5: 99.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right) \cdot normAngle, normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (*
    (fma 0.16666666666666666 n1_i (* (fma u -0.5 0.3333333333333333) n0_i))
    normAngle)
   normAngle
   (- n1_i n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf((fmaf(0.16666666666666666f, n1_i, (fmaf(u, -0.5f, 0.3333333333333333f) * n0_i)) * normAngle), normAngle, (n1_i - n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(Float32(fma(Float32(0.16666666666666666), n1_i, Float32(fma(u, Float32(-0.5), Float32(0.3333333333333333)) * n0_i)) * normAngle), normAngle, Float32(n1_i - n0_i)), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right), u, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, n1\_i, \mathsf{fma}\left(u, -0.5, 0.3333333333333333\right) \cdot n0\_i\right) \cdot normAngle, normAngle, n1\_i - n0\_i\right), u, n0\_i\right) \]
9. Add Preprocessing

Alternative 6: 99.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(normAngle \cdot normAngle\right), -2 \cdot n0\_i - n1\_i, n1\_i - n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (* -0.16666666666666666 (* normAngle normAngle))
   (- (* -2.0 n0_i) n1_i)
   (- n1_i n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf((-0.16666666666666666f * (normAngle * normAngle)), ((-2.0f * n0_i) - n1_i), (n1_i - n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(Float32(Float32(-0.16666666666666666) * Float32(normAngle * normAngle)), Float32(Float32(Float32(-2.0) * n0_i) - n1_i), Float32(n1_i - n0_i)), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot \mathsf{fma}\left(n1\_i, u \cdot \mathsf{fma}\left(u, u, -1\right), \left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto n0\_i + \color{blue}{u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{-1}{6} \cdot \left({normAngle}^{2} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right)\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(normAngle \cdot normAngle\right), -2 \cdot n0\_i - n1\_i, n1\_i - n0\_i\right), \color{blue}{u}, n0\_i\right) \]
9. Add Preprocessing

Alternative 7: 70.7% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot n0\_i\\ \mathbf{if}\;n0\_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 2.4999999206638063 \cdot 10^{-21}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) n0_i)))
   (if (<= n0_i -2.00000006274879e-22)
     t_0
     (if (<= n0_i 2.4999999206638063e-21) (* n1_i u) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (1.0f - u) * n0_i;
	float tmp;
	if (n0_i <= -2.00000006274879e-22f) {
		tmp = t_0;
	} else if (n0_i <= 2.4999999206638063e-21f) {
		tmp = n1_i * u;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 - u) * n0_i
    if (n0_i <= (-2.00000006274879e-22)) then
        tmp = t_0
    else if (n0_i <= 2.4999999206638063e-21) then
        tmp = n1_i * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(Float32(1.0) - u) * n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-2.00000006274879e-22))
		tmp = t_0;
	elseif (n0_i <= Float32(2.4999999206638063e-21))
		tmp = Float32(n1_i * u);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = (single(1.0) - u) * n0_i;
	tmp = single(0.0);
	if (n0_i <= single(-2.00000006274879e-22))
		tmp = t_0;
	elseif (n0_i <= single(2.4999999206638063e-21))
		tmp = n1_i * u;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.00000006e-22 or 2.49999992e-21 < n0_i

   1. Initial program 97.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Add Preprocessing

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0\_i} + n1\_i \cdot u \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, n1\_i \cdot u\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - u}, n0\_i, n1\_i \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

      5. lower-*.f32 98.4

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)} \]

   6. Taylor expanded in n0_i around inf

      \[\leadsto n0\_i \cdot \color{blue}{\left(1 - u\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.8%

      \[\leadsto \left(1 - u\right) \cdot \color{blue}{n0\_i} \]

   if -2.00000006e-22 < n0_i < 2.49999992e-21

   1. Initial program 95.0%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Add Preprocessing

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0\_i} + n1\_i \cdot u \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, n1\_i \cdot u\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - u}, n0\_i, n1\_i \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

      5. lower-*.f32 96.8

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)} \]

   6. Taylor expanded in n0_i around 0

      \[\leadsto n1\_i \cdot \color{blue}{u} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto u \cdot \color{blue}{n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.4999999206638063 \cdot 10^{-21}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.999999809593135 \cdot 10^{-20}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.4999999206638063 \cdot 10^{-21}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.999999809593135e-20)
   (* 1.0 n0_i)
   (if (<= n0_i 2.4999999206638063e-21) (* n1_i u) (* 1.0 n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.999999809593135e-20f) {
		tmp = 1.0f * n0_i;
	} else if (n0_i <= 2.4999999206638063e-21f) {
		tmp = n1_i * u;
	} else {
		tmp = 1.0f * n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-5.999999809593135e-20)) then
        tmp = 1.0e0 * n0_i
    else if (n0_i <= 2.4999999206638063e-21) then
        tmp = n1_i * u
    else
        tmp = 1.0e0 * n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-5.999999809593135e-20))
		tmp = Float32(Float32(1.0) * n0_i);
	elseif (n0_i <= Float32(2.4999999206638063e-21))
		tmp = Float32(n1_i * u);
	else
		tmp = Float32(Float32(1.0) * n0_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-5.999999809593135e-20))
		tmp = single(1.0) * n0_i;
	elseif (n0_i <= single(2.4999999206638063e-21))
		tmp = n1_i * u;
	else
		tmp = single(1.0) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.99999981e-20 or 2.49999992e-21 < n0_i

   1. Initial program 97.7%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Add Preprocessing

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0\_i} + n1\_i \cdot u \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, n1\_i \cdot u\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - u}, n0\_i, n1\_i \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

      5. lower-*.f32 99.2

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)} \]

   6. Taylor expanded in n0_i around inf

      \[\leadsto n0\_i \cdot \color{blue}{\left(1 - u\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

      \[\leadsto \left(1 - u\right) \cdot \color{blue}{n0\_i} \]

   9. Taylor expanded in u around 0

      \[\leadsto 1 \cdot n0\_i \]
   10. Step-by-step derivation

   11. Applied rewrites 60.3%

       \[\leadsto 1 \cdot n0\_i \]

   if -5.99999981e-20 < n0_i < 2.49999992e-21

   1. Initial program 95.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Add Preprocessing

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0\_i} + n1\_i \cdot u \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, n1\_i \cdot u\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - u}, n0\_i, n1\_i \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

      5. lower-*.f32 96.2

         \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

   5. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)} \]

   6. Taylor expanded in n0_i around 0

      \[\leadsto n1\_i \cdot \color{blue}{u} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.2%

      \[\leadsto u \cdot \color{blue}{n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.999999809593135 \cdot 10^{-20}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.4999999206638063 \cdot 10^{-21}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (fma (- n1_i n0_i) u n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((n1_i - n0_i), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(n1_i - n0_i), u, n0_i)
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \cdot u} + n0\_i \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), u, n0\_i\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot -0.5 - \cos normAngle \cdot \frac{normAngle}{\sin normAngle}, \frac{normAngle}{\sin normAngle} \cdot n1\_i\right), u, n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{n1\_i \cdot u + n0\_i \cdot \left(1 - u\right)} \]

   2. sub-neg N/A

      \[\leadsto n1\_i \cdot u + n0\_i \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(u\right)\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto n1\_i \cdot u + n0\_i \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u\right)\right) + 1\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto n1\_i \cdot u + \color{blue}{\left(\left(\mathsf{neg}\left(u\right)\right) \cdot n0\_i + 1 \cdot n0\_i\right)} \]

   5. *-lft-identity N/A

      \[\leadsto n1\_i \cdot u + \left(\left(\mathsf{neg}\left(u\right)\right) \cdot n0\_i + \color{blue}{n0\_i}\right) \]

   6. mul-1-neg N/A

      \[\leadsto n1\_i \cdot u + \left(\color{blue}{\left(-1 \cdot u\right)} \cdot n0\_i + n0\_i\right) \]

   7. *-commutative N/A

      \[\leadsto n1\_i \cdot u + \left(\color{blue}{\left(u \cdot -1\right)} \cdot n0\_i + n0\_i\right) \]

   8. associate-*r* N/A

      \[\leadsto n1\_i \cdot u + \left(\color{blue}{u \cdot \left(-1 \cdot n0\_i\right)} + n0\_i\right) \]

   9. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(n1\_i \cdot u + u \cdot \left(-1 \cdot n0\_i\right)\right) + n0\_i} \]

   10. *-commutative N/A

       \[\leadsto \left(\color{blue}{u \cdot n1\_i} + u \cdot \left(-1 \cdot n0\_i\right)\right) + n0\_i \]

   11. distribute-lft-in N/A

       \[\leadsto \color{blue}{u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} + n0\_i \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\left(n1\_i + -1 \cdot n0\_i\right) \cdot u} + n0\_i \]

   13. lower-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i + -1 \cdot n0\_i, u, n0\_i\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, u, n0\_i\right) \]

   15. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{n1\_i - n0\_i}, u, n0\_i\right) \]

   16. lower--.f32 97.9

       \[\leadsto \mathsf{fma}\left(\color{blue}{n1\_i - n0\_i}, u, n0\_i\right) \]

8. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)} \]
9. Add Preprocessing

Alternative 10: 39.2% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ n1\_i \cdot u \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (* n1_i u))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n1_i * u;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n1_i * u
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n1_i * u)
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n1_i * u;
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0\_i} + n1\_i \cdot u \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, n1\_i \cdot u\right)} \]

   3. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{1 - u}, n0\_i, n1\_i \cdot u\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

   5. lower-*.f32 97.7

      \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, \color{blue}{u \cdot n1\_i}\right) \]

5. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right)} \]

6. Taylor expanded in n0_i around 0

   \[\leadsto n1\_i \cdot \color{blue}{u} \]
7. Step-by-step derivation

8. Applied rewrites 41.2%

   \[\leadsto u \cdot \color{blue}{n1\_i} \]

9. Final simplification 41.2%

   \[\leadsto n1\_i \cdot u \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (normAngle u n0_i n1_i)
  :name "Curve intersection, scale width based on ribbon orientation"
  :precision binary32
  :pre (and (and (and (and (<= 0.0 normAngle) (<= normAngle (/ PI 2.0))) (and (<= -1.0 n0_i) (<= n0_i 1.0))) (and (<= -1.0 n1_i) (<= n1_i 1.0))) (and (<= 2.328306437e-10 u) (<= u 1.0)))
  (+ (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i) (* (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))) n1_i)))